Wednesday, July 10, 2013

"Is my green the same as yours?", revisited

Last May, I wrote an insightful and compelling blog post about whether we all see colors the same way. In my normal clever way, I called it "Is your green the same as my green?" Just a few days ago, I was given a bunch of really cool data that allows me to take another look at this question.


First, I need to tell you about Dr. Sophie Wuerger's recent paper, entitled "Colour Constancy Across the Life Span: Evidence for Compensatory Mechanisms". It is obvious that this is a scholarly paper because she spells "color" with a "u".
This technical paper is clearly above my head

Spoiler alert: I am about to give away the ending to her paper. If you don't want me to spoil a great story, then stop reading my blog. Go read her paper. Now.

It is well known that our vision changes as we age. For example, I used to enjoy lite beer. Today? I prefer to curl up with a darker beer. This is obviously due to the fact that the lenses in my eyes have slowly yellowed as I have aged [1]. For those of you are ardent fans of my blog, you already knew about the yellowing of the lens. I talked about that as one of the reasons that I may not see colors quite the same as someone who is still young enough to enjoy lite beer.

Well, Dr. W. completely spoiled that for me! Although the actual stimulus provided to the brain changes with each page of the calendar, Sophie's research shows that the brain corrects for the yellow sunglasses that we don in our old age.

So, in my last informative blog post on the subject, I concluded that the way we see color depends on how old we are. And then here comes along some research -- and research by a person who spells color with a u -- that says that somehow our perception of hue does not change that much as the lens yellows. I quote from the actual paper:

"Our main finding is that colour appearance mechanisms are to large extent unaffected by the known age-related changes in the optical media (yellowing of the lens) whereas the ability to discriminate between small colour differences is compromised with an increase in age."

Wow. Color me flummoxed. A yellowish shade of flummox.

Sophie's Choice
Here is the gist of her experiment. She sat 185 people (aged 18 to 75) in front of a computer monitor and displayed for them a selection of splendid colors [2]. The task before the experimental subjects was to select the color that was either the purest red, green, blue, or yellow in color. For example, a range of colors that were all kinda red in hue would be displayed. All of the colors would have roughly the same brightness and saturation, but would differ in hue. The simple question was phrased like this: "which of the colors has neither a yellowish or bluish shade?"

This test was performed with four color groups: red, green, yellow, and blue. (Yes, Billie, I can see your hand going up. You recognize those colors. I'll get to that later, ok?)  For each of the color groups (hues), there were nine different colors that were investigated.

In a previous blog post about organizing crayons, I introduced a fabulous diagram that helps even the mere novice understand color space. Each of the pages in this diagram represents a different hue. I have shown red, green, and blue, but there are of course a whole bunch of them. Maybe a hundred? Maybe a few hundred?
A fabulous diagram that helps even the mere novice understand color space

Let's just take a look at one of those pages (see blow). This page has all the colors that have a hue of red. Note that I have put nine "X" marks on this page. Those marks show roughly the nine "colors" (combinations of saturation and lightness) that were investigated.

Confused yet?  Let me summarize. There were four hues investigated (red, green, blue, and yellow). For each of these four hues, there were nine different investigations. Ok, so that makes investigations of 36 different colors.

Now, for each of 36 investigations, each of the guinea pigs [3] was asked to select from a zillion and a half different hues to find the one that most exemplified the trustworthy and faithful hue that we have come to know as "red". That was Sophie's Choice.

Here I have another view of the points in color space that were involved in this test. Here, we are looking at color space from above, so that the horizontal axis is the reddishness to greenishness, and the vertical axis is the yellowishness to bluishness. (Yes, Billie, those same four colors came up yet again. Hang on. I'll get to them. Yes, I know that something odd is going on.)
Really cool propeller-gram from the really cool experiment

Now I need to say one more thing about the diagram above. The points in the diagram are not "all the available choices". They are the colors that the experimental subjects actually chose as being red, green, blue, or yellow in hue.

Wow. You know... these results are pretty darn far flung. I'm not liking that. It kinda makes me think that we can't even agree on what red is! No wonder my gorgeous wife and I have trouble agreeing on color names as I reported in my post about spectrophotometric agreement!

But wait just a gol durned minute
The really cool propeller-gram showing the really cool results from a really cool experiment is also a really cool example of a misleading graph. I will zoom in on just one of the red investigations, the one with the highest saturation.
Looking at just one of the investigations

Each of the red dots represents an answer that at least one person (out of the 185) selected. In some cases - the cases where the dots are most far-flung - there was exactly one person so audacious as to say that this represents a true red hue. In the dots that are closer to the center of the line, there are literally zillions of people who chose that answer. The graph does a really lousy job of differentiating the popularity of the various answers [4].

The graph below is a less distorted way of looking at the same data. This is a histogram, showing that there is a pretty tight clustering. The labels are hard to read unless you double click on the image, but about half of the subjects selected a hue angle between 16 and 18 degrees. I am going to call that pretty darn tight, and make the declaration that we all can agree on at least the hue for "red".
Histogram of hue angle for one of the red investigations

What do red, green, yellow, and green mean?
And now for the big moment. Finally. A definition of these four hues!
Note: The uncertainty column was determined by computing the standard deviation of the hue angles, and dividing this by the square root of the number of observations. This is an estimate of how close the average is to the "real" value.

Note: The table above is just the averages for one investigation of each of the four hues. I have not yet looked at all nine. Also, Sophie has provided me with gobs more data. This is only the "dark adapted" data.

Something odd is going on with my Hering
Billie, now to your question. First you noticed that the four hues that Dr. Wuerger chose were red, green, yellow and blue, and that these are the colors of the four chromatic directions in CIELAB space. Very good, Billie. You have been paying attention.

This whole idea of the names for the four directions goes back to a fellow by the name of Ewald Hering. He came up with the idea that we perceive color based on three sets of opponents. Colors are either reddish or greenish. Colors are either yellowish or bluish. Colors are either light or dark. Any color can be defined by where it rates in each of those three attributes.

Noted color scientist Ewald Hering, and noted food of color scientists

Now Billie, you had another question? Yes... you looked at the graph labelled "really cool propeller-gram from a really cool experiment" and started thinking about Hering's theory? What really cool thing did you notice about the graph? OH?

Green is not diametrically opposed to red. To be precise, they are not 180 degrees apart, but about 143.  And blue and yellow are not either. Yellow and blue are about 120 degrees apart.

Hering was wrong? Heresy. Pure blasphemy. I should clarify... The theory is correct qualitatively. But to be quantitatively correct, you need to adjust the meaning of the words. "Pure red" and a certain shade of bluish green (maybe cyan?) are opponents. And "pure yellow" and a certain shade of reddish blue(maybe violet?) are opponents.

This is work in progress. I have just started analyzing the data that Dr. Wuerger has graciously shared with me. But let me tell you one thing... If I ever invent a color space of my own, red is going to lie right on the horizontal axis, and not some silly 16 degrees off.  And I am going to try to get yellow right on the vertical axis. And I am going to try to resolve the issue of what the true opponents of these colors are.

Oh... and my new space is going to be perceptually linear, gosh darn it. None of those ugly deltaE 2000 equations. And I will call the axes JMG.

-John the Math Guy

[1] My predilection for the darker and more full bodied beers is well known. This is often cited in the technical journals as further proof of the yellowing of the lens. There has also been considerable speculation about whether this penchant for dark and full bodied extends from beer to my taste in women. I decline to comment on this controversy. I think it's just silly.

[2] Think of that. 185 people. In front of one computer monitor. She must have the most awesome huge big screen TV in her neighborhood! I know where I am going to watch the next Academy Awards... Wait... they weren't all in front of the monitor at the same time? Oh.  Ok, so I am still open to invitations to watch the Academy Awards.

[3] I have been assured that no animals were harmed in the filming of this experiment.

[4] This was meant as a didactic warning about interpreting scatter plots when the number of points is way big. Basically, the more data points, the larger the impression of the variation. Some day I will do blog post about that cuz I have something cool to show off. It involves using singular value decomposition to compute the square root of a co-variance matrix, so you know that you will need to check back to read that post!


  1. John, there is one other pissible explanation to your unique hue dilemma. Perhaps the spacing of hue in CEILAB is not uniform. Look for articles a space caled IPT and see what thosevauthors have to say about unique hues.

  2. Thanks for the tip... I am reading some papers on IPT.